Did you know that the Hypergeometric Distribution is hugely similar to the Binomial Distribution?

Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Their differences lie in the way that sampling is done.
The binomial distribution has a fixed number of independent trials, whereas the hypergeometric distribution has a set number of dependent trials.
In other words, we will use the hypergeometric distribution whenever we have sampling without replacement!
Additionally, this distribution enables us to deal with situations arising when we sample from batches with a known number of defective items.
Formulas
Suppose a set of N objects contains k objects that are classified as successes and N-k objects that are classified as failures, then with a sample size of n randomly selected objects, without replacement. In that case, the Hypergeometric random variable X has the following properties as noted by Penn State.

Hypergeometric Distribution (PMF, Mean and Variance)
Now I know these formulas look intimidating but trust me when I tell you that they’re easy to use and understand when you see them in action.
Worked Example
For example, suppose Harry Potter and Ron Weasley are on the Hogwarts Express, and Harry buys a box of Bertie Bott’s every flavor bean from the lunch trolley (complements of JK Rowling).
Let’s assume this box contains ten beans of the following flavors:
- Banana
- Blueberry
- Booger
- Earwax
- Grass
- Green Apple
- Marshmallow
- Rotten Egg
- Lemon
- Vomit
This means there are six tasty flavors and four yucky flavors (i.e., booger, earwax, rotten egg, and vomit). Harry decides to share the box with his friend Ron. So, Ron reaches in and randomly selects five beans and eats them.
What is the chance that Ron chooses the two yucky flavors?

Hypergeometric Probability Example
This means that Ron has a 0.476 chance of choosing two yucky flavors from a sample size of 5 beans, knowing that there were four yucky flavors in the box of 10!
And did you know that we can extend these ideas to more than just two choices?
The Multivariate Hypergeometric Distribution states that

Multivariate Hypergeometric Distribution PMF
Again, this formula can look a bit scary, but it’s straightforward to use in practice.
Worked Example
So, let’s look at an example!
Suppose we want to find the probability that a committee of 10 people chosen from a group consisting of 40 principals, 35 teachers, and 25 students, will include three principals, five teachers, and two students.

Multivariate Hypergeometric Example
This means there is a 0.0556 chance that precisely 3 principals, five teachers, and two students will be chosen for the committee.
This lesson will walk you through detailed examples of how to recognize the hypergeometric distribution and how to apply the formulas for probability, expectancy, and variance without getting lost or confused.
Let’s jump right in!
Hypergeometric Distribution – Lesson & Examples (Video)
51 min
- Introduction to Video: Hypergeometric Distribution
- 00:00:41 – Overview of the Hypergeometric Distribution and formulas
- Exclusive Content for Members Only
- 00:12:21 – Determine the probability, expectation and variance for the sample (Examples #1-2)
- 00:26:08 – Find the probability and expected value for the sample (Examples #3-4)
- 00:35:50 – Find the cumulative probability distribution (Example #5)
- 00:46:33 – Overview of Multivariate Hypergeometric Distribution with Example #6
- Practice Problems with Step-by-Step Solutions
- Chapter Tests with Video Solutions
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